Linear Operators: General theory |
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Page 228
... derivatives of f , of arbitrarily large order converge to the corresponding partial derivatives of f uniformly in V. From this theorem , we can easily derive the useful fact that if ƒ is analytic in the Cartesian product U1X ... X × Un ...
... derivatives of f , of arbitrarily large order converge to the corresponding partial derivatives of f uniformly in V. From this theorem , we can easily derive the useful fact that if ƒ is analytic in the Cartesian product U1X ... X × Un ...
Page 384
... derivatives including those of order n converge uniformly to the corresponding derivatives of f . ) Of the many extensions of the Weierstrass theorem for [ 0 , 1 ] we mention the striking theorem of Müntz [ 1 ] which asserts that the ...
... derivatives including those of order n converge uniformly to the corresponding derivatives of f . ) Of the many extensions of the Weierstrass theorem for [ 0 , 1 ] we mention the striking theorem of Müntz [ 1 ] which asserts that the ...
Page 655
... derivatives f ' ' lying in L ,, and which is defined by Aff " for fe D ( A ) . 14 Let X = L ( -∞ , ∞ ) , 1 ≤ p ... derivatives and with second derivatives f " lying in L ,, and which is defined by Af = f ' ' , ƒ € D ( A ) . 16 Let X ...
... derivatives f ' ' lying in L ,, and which is defined by Aff " for fe D ( A ) . 14 Let X = L ( -∞ , ∞ ) , 1 ≤ p ... derivatives and with second derivatives f " lying in L ,, and which is defined by Af = f ' ' , ƒ € D ( A ) . 16 Let X ...
Contents
A Settheoretic Preliminaries | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets Cauchy sequence compact operator complex numbers contains continuous functions continuous linear converges convex set Corollary countably additive DEFINITION denote dense differential equations disjoint Doklady Akad domain E₁ element exists f₁ finite dimensional finite number function defined function f Hausdorff space Hence Hilbert space homeomorphism inequality integral L₁ L₁(S Lebesgue Lemma Let f linear functional linear map linear operator linear topological space measurable function measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field open set operator topology positive measure space Proc PROOF proved real numbers Riesz Russian S₁ scalar semi-group sequentially compact Show spectral strong operator topology subset subspace Suppose T₁ theory topological space u-integrable u-measurable uniformly unit sphere valued function weakly compact zero ΕΕΣ